Volume Radius Problem: Solving with Cylindrical Method

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Homework Help Overview

The discussion revolves around calculating the volume of a region defined by the curve y = x^2, the x-axis, and vertical lines at x = -2 and x = -1, when revolved around the y-axis using the cylindrical method.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the assumptions regarding the radius and height of the cylindrical shells, with some questioning the correct limits and the definition of the radius in relation to the axis of rotation.

Discussion Status

There is an ongoing examination of the assumptions about the radius and limits of integration. Some participants affirm the validity of using the first quadrant for calculations, while others seek clarification on the definition of the radius in this context.

Contextual Notes

Participants are discussing the implications of revolving the region around the y-axis and the corresponding definitions of radius and height in the cylindrical method. There is a mention of a broken link that was intended to provide additional resources.

frozen7
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The volume surrounded by y = x^2 , y=0, x=-2, x=-1 , revolved about y axis,by using cylindrical method.

The region is actually within x=-1 to -2 and it can actually same with x=1 to 2.

radius = 2-x
height = x^2

Is my assumption correct?
 
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Your radius should simply be x. With limits of: x = 1 to 2

You assumption about being able to use the region in the 1st quadrant is correct, yes.
 
How come the radius should be x?
 
The radius should be x because the radius of the cylinder is the distance from the origin.
The volume of revolution is about the y-axis, which passes through the origin. And you are finding the volumes of all the thin cylinders starting with cylinders with a radius of x = 1 to x= 2.

See here for a further treatment of the cylindrical method.
 
Last edited:
Something wrong with the link you have given.
 
Sorry 'bout that. Forgot to add in the url.
The link is fixed now.
 

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